What is the difference between a Maclaurin series and a Taylor series?

Conditions

• The Taylor formula is taken with center a.
• Setting $a=0$ recovers the displayed Maclaurin formula.

Reasoning, step by step

1. Identify the Maclaurin series definition: $f(x)$ = sum $c_n x^n$ with $c_n = f^{(n)}(0) / n$!.
2. Identify the Taylor series definition: $f(x)$ = sum $c_n (x-a)^n$ with $c_n = f^{(n)}(a) / n$!.
3. Compare the two formulas to see that replacing the center 0 with an arbitrary point a shifts the powers from $x^n$ to (x-a)^n and the derivative evaluation from 0 to a.
4. Conclude that the Maclaurin series is simply the Taylor series with $a=0$.

Example

The speaker says, "A Maclaurin series where a is zero is just a special case of the true thing, Taylor series." The two boxed definitions differ only by replacing 0 with a in the center, the power, the convergence condition, and the derivative evaluation point.

Common misconceptions

• Thinking the choice of 0 is mathematically essential and that the coefficient-extraction method only works for series centered at 0.
• Believing that Taylor series and Maclaurin series are completely unrelated concepts rather than a generalization and a special case.

Watch the explanation

Explore next

• How does the video begin computing the Maclaurin series for $e^x$?
• What is the coefficient formula for a Maclaurin series centered at 0?